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In this thesis we develop an explanation and proof of the Hasse-Minkowski Theorem for homogeneous quadratic forms in two and three variables using only undergraduate number theory. The goal of this approach is to provide an accessible introduction to this celebrated result of number theory for undergraduates and advanced high school students. Our account of the Hasse-Minkowski Theorem will be expository in nature, providing the reader with the necessary background information to state and prove the theorem. However, some mathematical maturity of the reader is presumed, including familiarity with unique prime factorization, greatest common divisor, least common multiple, the Division Algorithm, the Euclidean Algorithm, the Pigeonhole Principle, and some other basic notions from elementary number theory, set theory, and linear algebra. The remainder of the information necessary for the explanation and proof of the Hasse-Minkowski Theorem is provided in the paper, making it largely self-contained in this respect. Moreover, throughout the paper we provide the reader with several numerical examples of the various concepts and methods introduced